[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 12

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| If W is a submanifold of a manifold Y of
| class C^p (p >= 1), then the inclusion:
|
| i : W -> Y
|
| induces a map:
|
| T_w i  :  T_w (W) -> T_w (Y)
|
| which is in fact an injection.
|
| From the definition of a submanifold, one sees immediately
| that the image of T_w i splits.  It will be convenient to
| identify T_w (W) in T_w (Y) if no confusion can result.
|
| A morphism f : X -> Y will be said to be "transversal"
| over the submanifold W of Y if the following condition
| is satisfied.
|
| Let x in X be such that f(x) is in W.
| Let (V, r) be a chart at f(x) such that
| r : V -> V_1 x V_2 is an isomorphism on
| a product, with:
|
| r(f(x)) = (0, 0)  and  r(W |^| V) = V_1 x 0.
|
| Then there exists an open neighborhood U of x
| such that the composite map:
|
|      f         r               proj
| U ------> V ------> V_1 x V_2 ------> V_2
|
| is a submersion.
|
| [Here, "proj" denotes the "projection" proj : V_1 x V_2 -> V_2.]
|
| In particular, if f is transversal over W, then
| f^(-1) (W) is a submanifold of X, because the
| inverse image of 0 by our local composite map:
|
| proj o r o f
|
| is equal to the inverse image of W |^| V by r.
|
| Lang, DARM, p. 27.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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